1. Field of the Invention
The present invention relates generally to differential scanning calorimeters (DSCs), and more specifically to methods of accounting for heat leakage in DSCs.
2. Background Information
A DSC is a symmetrical instrument, comprising a sample and reference calorimeter within a common thermal enclosure, where the two calorimeters are intended to be identical. DSCs have a sensor which measures the temperature difference between the sample and the reference position in the respective calorimeters. A sample to be analyzed is loaded into a pan and placed on the sample position of the sensor and an inert reference material is loaded into a pan and placed on the reference position of the sensor (alternatively, an empty pan is often used as the reference). The sensor is installed in an oven whose temperature is varied dynamically according to a desired temperature program. The temperature program for conventional DSCs typically includes combinations of linear temperature ramps and constant temperature segments. Modulated DSCs use a temperature program in which periodic temperature modulations are superimposed on linear ramps. Modulated DSCs are described in U.S. Pat. No. 5,224,775, which is incorporated by reference herein.
Conventional DSC Heat Flow Rate Measurement
In conventional DSCs, the measured quantities are the difference between the temperatures of the sample and reference sensors and the sample sensor temperature. During the dynamic portion of the DSC experiment, a differential temperature is created between the sample and reference positions on the sensor. The temperature difference is the result of the difference between the heat flow to the sample and the heat flow to the reference. The temperature difference is assumed to be proportional to the difference in heat flow to the sample as compared to the heat flow to the reference, thus the differential temperature may be used to measure the heat flow to the sample using the equationq=E(T)ΔT
Where, E(T) is a temperature dependent factor that reflects the proportionality of the measured differential temperature to the sample heat flow rate. A fundamental underlying assumption is that the sample and reference sides of the calorimeter are perfectly symmetrical. In reality the balance is less than perfect which is reflected in the observation that when the calorimeter is operated empty, the resulting heat flow rate signal is not zero as it should be if the DSC were symmetrical as assumed. Additionally, it is well known that the heat flow rate signal is smeared, as a result of the heat capacity of the sensor and pans and the differences in heating rate that exist between the sample and reference. The beating rate differences exist in conventional DSC during a transition or generally during a MDSC experiment. However, those shortcomings may be overcome by using the DSC apparatus and heat flow rate measurement methods disclosed below.
DSC Heat Flow Rate Measurement Including Sensor Asymmetry
FIG. 1 shows a thermal network model that may be used to represent heat flux in certain DSC sensors. To is the temperature at the base of the sensor near its connection to the oven, Ts is the temperature of the sample position of the sensor and Tr is the temperature of the reference position of the sensor. Rs and Rr represent the thermal resistance between the base of the sensor and the sample and reference positions, respectively. Cs and Cr represent the thermal capacitance of the sample and reference portions of the sensor. Thermal capacitance is the product of mass and specific heat and is a measure of the heat storage capacity of a body, i.e., it is the heat capacity of the body. The heat flow rate to the sample and the reference are qs and qr, respectively. It should be understood that qs and qr include heat flow to sample and reference pans. During the execution of a thermal program, the base temperature of the sensor To follows the thermal program. The temperatures at the sample and reference positions, Ts and Tr, lag the base temperature To due to heat flowing to the sample and to the reference and heat which is stored within the sensor in sensor sample thermal capacitance Cs and sensor reference thermal capacitance Cr.
Performing a heat flow balance on the sample side of the sensor yields the measured sample heat flow rate
      q    s    =                              T          o                -                  T          s                            R        s              -                  C        s            ·                        ⅆ                      T            s                                    ⅆ          τ                    which is equal to the heat flow rate through the sensor sample thermal resistance Rs minus the heat stored in Cs. Similarly, a heat balance on the reference side of the sensor gives the measured reference heat flow rate
      q    r    =                              T          o                -                  T          r                            R        r              -                  C        r            ·                        ⅆ                      T            r                                    ⅆ          τ                    which is equal to the heat flow rate through sensor reference thermal resistance Rr minus the heat stored in Cr. In the equations herein, τ represents time.
The desired quantity (the differential heat flow to the sample with respect to the reference) is the difference between the sample and reference heat flows:q=qs−qrSubstituting for qs and qr yields:
  q  =                              T          o                -                  T          s                            R        s              -                  C        s            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                            T          o                -                  T          r                            R        r              +                  C        r            ·                        ⅆ                      T            r                                    ⅆ          τ                    Substituting the following expressions for two temperature differences in the differential scanning calorimeter,ΔT=Ts−TrΔTo=To−Tswhere ΔT is the temperature difference between the sensor sample and the reference positions and ΔTo is the temperature difference between the sample position and a position at the base of the sensor, results in the DSC heat flow equation:
  q  =            Δ      ⁢                          ⁢                        T          o                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                                  ⁢        T                    R        r              +                  (                              C            r                    -                      C            s                          )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                  C        r            ·                                    ⅆ            Δ                    ⁢                                          ⁢          T                          ⅆ          τ                    
The DSC heat flow equation has four terms. The first term accounts for the effect of the difference between the sensor sample thermal resistance and the sensor reference thermal resistance. The second term is the conventional DSC heat flow. The third term accounts for the effect of the difference between the sensor sample thermal capacitance and the sensor reference thermal capacitance. The fourth term accounts for the effect of the difference between the heating rates of the sample and reference sides of the DSC. Conventionally, when this equation is applied to the DSC heat flow, the first and third terms are zero because Rs and Rr are assumed to be equal and Cs and Cr are also assumed to be equal. The fourth term is zero because the heating rate difference between the sample and reference is ignored.
Heat Leakage Effects
Heat leakage is defined as heat that flows between the sample or reference and the enclosure without being measured. This leakage, or unmeasured heat flow, contributes to the uncertainty of experiments in which DSCs are utilized.
Advantages that are commonly associated with using twin calorimeters in DSCs as described above include cancellation of heat leakage and temperature disturbances common to both calorimeters. If the calorimeters each contain empty sample pans, and are heated at the same rate, there should be no difference between the sensor temperatures of each enclosure. However, in practice, the two calorimeters will always generally measure a different heat flow rate. Even if the calorimeters are perfectly matched, the presence of a sample causes the sample and reference heat flow rates to be imbalanced.
When a DSC is not balanced, the leakage heat flows will not be balanced and will not cancel when the difference is taken between the measured sample and reference heat flow rates, which contributes to a measurement error in the heat flow rate. In general, the leakage heat flows are a small fraction of the measured heat flow. If the imbalance is small, the measurement error is small, as a difference between two relatively small quantities. The magnitude of the errors has been estimated for disk-type heat flux DSCs at between 1 and 5% by numerical simulation.
In principle, the uncertainty can be reduced by calibration of the heat flow rate. However, experimental conditions that are different from the calibration conditions will cause uncertainty due to heat leakage to increase. Differences in sample heat capacity and differences in the transitions within a sample are the primary differences between calibration and experimental conditions, because the experimental sample in general will have a different heat capacity from the calibrant and will have different transitions.
A manifestation of the effect of heat leakage that is well known in differential scanning calorimetry is the difference between heat flow calibration and peak area calibration. In heat flow calibration, a sample having known specific heat capacity, such as sapphire, is heated at a constant heating rate and the heat flow rate is measured. Heat capacity of the sample is obtained by dividing the heat flow rate signal by the heating rate, giving the measured heat capacity. The ratio of the actual value to the measured value gives the heat flow calibration factor for the range of temperatures of the experiment. It is applied to the results of subsequent experiments to improve the accuracy of the heat flow measurement. In peak area calibration, the sample is a material with a transition whose enthalpy is well known and highly repeatable, such as the melt of a pure metal like indium. A calibration experiment is performed and the area of the resulting heat flow rate peak is measured by integrating the heat flow signal versus time over an appropriate baseline. The area of the peak provides the measured enthalpy of the transition. The ratio of the measured enthalpy to the actual enthalpy is the peak area calibration factor.
In principle, at the temperature of the transition, the peak area and the heat flow calibration factors should be identical, or very nearly so, but in general the agreement is rather poor. The reason for this disagreement is the large temperature imbalance created by the transition and the attendant increase in sample heat leakage due to the imbalance. For example, when calibrating with a typical indium sample of 5 mg at 10° C. min, the temperature difference between the sample and the reference before the melt is of the order of only a few tenths of 1° C., but that increases to 1° C. or more at the peak of the transition. Thus, the temperatures of the sample and reference calorimeters are very far from being identical, the twin principle is severely violated, and the leakage heat flows will be much different and will not cancel each other out.
Another example of the effect of heat leakage is the well known effect of thermal conductivity of the purge gas on DSC cell calibration. In either heat flow rate or peak area calibration, it is well known that changing the purge gas causes both the peak area and the heat flow rate calibration to change. In the absence of heat leakage, both the heat flow rate calibration and peak area calibration would depend only upon thermal resistance of the calorimeters and would be independent of the purge gas used.